Quantum Query Complexity of Finding a Tarski Fixed Point on a High-Dimensional Grid

2026-09-03

AI Summary

研究使用非负谱对手方法证明了在高维网格上找到Tarski不动点的量子查询复杂性的下界，提出了树-过滤对手方法。

Abstract

The Knaster-Tarski fixed-point theorem states that every monotone function over a complete lattice has a fixed point. Beyond its fundamental role in order theory, the theorem and its algorithmic variants have found broad applications in areas such as economics, game theory, and programming languages. While the query complexity of finding a Tarski fixed point has been extensively studied in classical models, comparatively little is known in the quantum setting. We prove an $Ω(k\log n)$ quantum query lower bound for finding a fixed point of a monotone function on $[n]^k$, using the nonnegative spectral adversary method. In the two extremal regimes $n = 2$ and $k = 1$, our quantum lower bound matches the previous classical lower bounds $Ω(k)$ and $Ω(\log n)$, respectively. For $n, k\geq 2$, our bound improves the best previous classical lower bound when $n < k$ and is within a factor of $\log n / \log k$ compared to the known classical lower bound when $n \geq k$. To construct the adversary matrix, we develop the Tree--Filtration Adversary Method. Besides yielding our lower bound, the method offers a more transparent combinatorial interpretation of the nonnegative spectral adversary method. When the hard instances of a problem admit a tree-like organization and suggest an intuition analogous to classical decision-tree lower bounds, our method provide a promising approach to establishing quantum complexity lower bounds.

Problem

• Quantum Query Complexity
• Tarski Fixed Point
• High-Dimensional Grid

Innovation

• Quantum Query Complexity
• Tarski Fixed Point
• Tree--Filtration Adversary Method
• Nonnegative Spectral Adversary